Shift operators and Rodrigues-type formulas: Difference between revisions

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Shift operators and Rodrigues-type formulas

f ′ ⁢ ( x ) = d d ⁢ x ⁢ f ⁢ ( x ) = d ⁢ f d ⁢ x ⁢ ( x ) = d ⁢ f ⁢ ( x ) d ⁢ x superscript 𝑓 ′ 𝑥 𝑑 𝑑 𝑥 𝑓 𝑥 𝑑 𝑓 𝑑 𝑥 𝑥 𝑑 𝑓 𝑥 𝑑 𝑥 {\displaystyle{\displaystyle{\displaystyle f^{\prime}(x)=\frac{d}{dx}f(x)=% \frac{df}{dx}(x)=\frac{df(x)}{dx}}}} {\displaystyle f'(x)=\frac{d}{dx}f(x)=\frac{df}{dx}(x)=\frac{df(x)}{dx} }
Δ ⁢ f ⁢ ( x ) = f ⁢ ( x + 1 ) - f ⁢ ( x ) Δ 𝑓 𝑥 𝑓 𝑥 1 𝑓 𝑥 {\displaystyle{\displaystyle{\displaystyle\Delta f(x)=f(x+1)-f(x)}}} {\displaystyle \Delta f(x)=f(x+1)-f(x) }
∇ ⁡ f ⁢ ( x ) = f ⁢ ( x ) - f ⁢ ( x - 1 ) ∇ 𝑓 𝑥 𝑓 𝑥 𝑓 𝑥 1 {\displaystyle{\displaystyle{\displaystyle\nabla f(x)=f(x)-f(x-1)}}} {\displaystyle \nabla f(x)=f(x)-f(x-1) }
δ ⁢ f ⁢ ( x ) = f ⁢ ( x + 1 2 ⁢ i ) - f ⁢ ( x - 1 2 ⁢ i ) 𝛿 𝑓 𝑥 𝑓 𝑥 1 2 imaginary-unit 𝑓 𝑥 1 2 imaginary-unit {\displaystyle{\displaystyle{\displaystyle\delta f(x)=f(x+\textstyle\frac{1}{2% }\mathrm{i})-f(x-\textstyle\frac{1}{2}\mathrm{i})}}} {\displaystyle \delta f(x)=f(x+\textstyle\frac{1}{2}\iunit)-f(x-\textstyle\frac{1}{2}\iunit) }
δ 2 ⁢ f ⁢ ( x ) = f ⁢ ( x + i ) - f ⁢ ( x ) - f ⁢ ( x ) + f ⁢ ( x - i ) = f ⁢ ( x + i ) - 2 ⁢ f ⁢ ( x ) + f ⁢ ( x - i ) superscript 𝛿 2 𝑓 𝑥 𝑓 𝑥 imaginary-unit 𝑓 𝑥 𝑓 𝑥 𝑓 𝑥 imaginary-unit 𝑓 𝑥 imaginary-unit 2 𝑓 𝑥 𝑓 𝑥 imaginary-unit {\displaystyle{\displaystyle{\displaystyle\delta^{2}f(x)=f(x+\mathrm{i})-f(x)-% f(x)+f(x-\mathrm{i})=f(x+\mathrm{i})-2f(x)+f(x-\mathrm{i})}}} {\displaystyle \delta^2 f(x)=f(x+\iunit)-f(x)-f(x)+f(x-\iunit)=f(x+\iunit)-2f(x)+f(x-\iunit) }
δ ⁢ x = x + 1 2 ⁢ i - ( x - 1 2 ⁢ i ) = i   and   δ ⁢ x 2 = ( x + 1 2 ⁢ i ) 2 - ( x - 1 2 ⁢ i ) 2 = 2 ⁢ i ⁢ x formulae-sequence 𝛿 𝑥 𝑥 1 2 imaginary-unit 𝑥 1 2 imaginary-unit imaginary-unit and 𝛿 superscript 𝑥 2 superscript 𝑥 1 2 imaginary-unit 2 superscript 𝑥 1 2 imaginary-unit 2 2 imaginary-unit 𝑥 {\displaystyle{\displaystyle{\displaystyle\delta x=x+\textstyle\frac{1}{2}% \mathrm{i}-(x-\textstyle\frac{1}{2}\mathrm{i})=\mathrm{i}\quad\textrm{and}% \quad\delta x^{2}=(x+\textstyle\frac{1}{2}\mathrm{i})^{2}-(x-\textstyle\frac{1% }{2}\mathrm{i})^{2}=2\mathrm{i}x}}} {\displaystyle \delta x=x+\textstyle\frac{1}{2}\iunit-(x-\textstyle\frac{1}{2}\iunit)=\iunit \quad\textrm{and}\quad \delta x^2=(x+\textstyle\frac{1}{2}\iunit)^2-(x-\textstyle\frac{1}{2}\iunit)^2=2\iunit x }
Δ ⁢ λ ⁢ ( x ) = 2 ⁢ x + γ + δ + 2   and   ∇ ⁡ λ ⁢ ( x ) = 2 ⁢ x + γ + δ formulae-sequence Δ 𝜆 𝑥 2 𝑥 𝛾 𝛿 2 and ∇ 𝜆 𝑥 2 𝑥 𝛾 𝛿 {\displaystyle{\displaystyle{\displaystyle\Delta\lambda(x)=2x+\gamma+\delta+2% \quad\textrm{and}\quad\nabla\lambda(x)=2x+\gamma+\delta}}} {\displaystyle \Delta\lambda(x)=2x+\gamma+\delta+2\quad\textrm{and}\quad\nabla\lambda(x)=2x+\gamma+\delta }
Δ ⁢ μ ⁢ ( x ) = q - x - 1 ⁢ ( 1 - q ) ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x + 2 )   and   ∇ ⁡ μ ⁢ ( x ) = q - x ⁢ ( 1 - q ) ⁢ ( 1 - γ ⁢ δ ⁢ q 2 ⁢ x ) formulae-sequence Δ 𝜇 𝑥 superscript 𝑞 𝑥 1 1 𝑞 1 𝛾 𝛿 superscript 𝑞 2 𝑥 2 and ∇ 𝜇 𝑥 superscript 𝑞 𝑥 1 𝑞 1 𝛾 𝛿 superscript 𝑞 2 𝑥 {\displaystyle{\displaystyle{\displaystyle\Delta\mu(x)=q^{-x-1}(1-q)(1-\gamma% \delta q^{2x+2})\quad\textrm{and}\quad\nabla\mu(x)=q^{-x}(1-q)(1-\gamma\delta q% ^{2x})}}} {\displaystyle \Delta\mu(x)=q^{-x-1}(1-q)(1-\gamma\delta q^{2x+2})\quad\textrm{and}\quad \nabla\mu(x)=q^{-x}(1-q)(1-\gamma\delta q^{2x}) }
Δ ⁢ λ ⁢ ( x ) = q - x - 1 ⁢ ( 1 - q ) ⁢ ( 1 - c ⁢ q 2 ⁢ x - N + 1 )   and   ∇ ⁡ λ ⁢ ( x ) = q - x ⁢ ( 1 - q ) ⁢ ( 1 - c ⁢ q 2 ⁢ x - N - 1 ) formulae-sequence Δ 𝜆 𝑥 superscript 𝑞 𝑥 1 1 𝑞 1 𝑐 superscript 𝑞 2 𝑥 𝑁 1 and ∇ 𝜆 𝑥 superscript 𝑞 𝑥 1 𝑞 1 𝑐 superscript 𝑞 2 𝑥 𝑁 1 {\displaystyle{\displaystyle{\displaystyle\Delta\lambda(x)=q^{-x-1}(1-q)(1-cq^% {2x-N+1})\quad\textrm{and}\quad\nabla\lambda(x)=q^{-x}(1-q)(1-cq^{2x-N-1})}}} {\displaystyle \Delta\lambda(x)=q^{-x-1}(1-q)(1-cq^{2x-N+1})\quad\textrm{and}\quad \nabla\lambda(x)=q^{-x}(1-q)(1-cq^{2x-N-1}) }
Δ ⁢ q - x = q - x - 1 ⁢ ( 1 - q )   and   ∇ ⁡ q - x = q - x ⁢ ( 1 - q ) formulae-sequence Δ superscript 𝑞 𝑥 superscript 𝑞 𝑥 1 1 𝑞 and ∇ superscript 𝑞 𝑥 superscript 𝑞 𝑥 1 𝑞 {\displaystyle{\displaystyle{\displaystyle\Delta q^{-x}=q^{-x-1}(1-q)\quad% \textrm{and}\quad\nabla q^{-x}=q^{-x}(1-q)}}} {\displaystyle \Delta q^{-x}=q^{-x-1}(1-q)\quad\textrm{and}\quad\nabla q^{-x}=q^{-x}(1-q) }
D q ⁢ f ⁢ ( x ) := δ q ⁢ f ⁢ ( x ) δ q ⁢ x assign subscript 𝐷 𝑞 𝑓 𝑥 subscript 𝛿 𝑞 𝑓 𝑥 subscript 𝛿 𝑞 𝑥 {\displaystyle{\displaystyle{\displaystyle D_{q}f(x):=\frac{\delta_{q}f(x)}{% \delta_{q}x}}}} {\displaystyle D_qf(x):=\frac{\delta_qf(x)}{\delta_qx} }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


δ q ⁢ f ⁢ ( e i ⁢ θ ) = f ⁢ ( q 1 2 ⁢ e i ⁢ θ ) - f ⁢ ( q - 1 2 ⁢ e i ⁢ θ ) subscript 𝛿 𝑞 𝑓 imaginary-unit 𝜃 𝑓 superscript 𝑞 1 2 imaginary-unit 𝜃 𝑓 superscript 𝑞 1 2 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle\delta_{q}f({\mathrm{e}^{\mathrm{i}% \theta}})=f(q^{\frac{1}{2}}{\mathrm{e}^{\mathrm{i}\theta}})-f(q^{-\frac{1}{2}}% {\mathrm{e}^{\mathrm{i}\theta}})}}} {\displaystyle \delta_qf(\expe^{\iunit\theta}) =f(q^{\frac{1}{2}}\expe^{\iunit\theta})-f(q^{-\frac{1}{2}}\expe^{\iunit\theta}) }
δ q ⁢ x = - 1 2 ⁢ q - 1 2 ⁢ ( 1 - q ) ⁢ ( e i ⁢ θ - e - i ⁢ θ ) subscript 𝛿 𝑞 𝑥 1 2 superscript 𝑞 1 2 1 𝑞 imaginary-unit 𝜃 imaginary-unit 𝜃 {\displaystyle{\displaystyle{\displaystyle\delta_{q}x=-\textstyle\frac{1}{2}q^% {-\frac{1}{2}}(1-q)({\mathrm{e}^{\mathrm{i}\theta}}-{\mathrm{e}^{-\mathrm{i}% \theta}})}}} {\displaystyle \delta_qx=-\textstyle\frac{1}{2}q^{-\frac{1}{2}}(1-q)(\expe^{\iunit\theta}-\expe^{-\iunit\theta}) }

Substitution(s): x = cos ⁡ θ 𝑥 𝜃 {\displaystyle{\displaystyle{\displaystyle x=\cos\theta}}}


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